The Car and the Goats

You are a contestant on a television game show.
Before you are three closed doors. One of them hides a car, which you
want to win; the other two hide goats (which you do not want to win).

You get to pick one of the doors, and you will win what is behind it.

However, the way the game works is that the door you pick does not get
opened immediately. Instead, the host (Monty Hall)
will open one of the other doors
to reveal a goat. He will then give you a chance to change your mind:
you can switch and pick the other closed door instead, or stay
with your original choice.

Which of these two strategies gives you the better chance of winning the
car? This simple question recently caused quite a storm of
mathematical controversy!

Why don't you
play the game a few times
to get a sense of which strategy is better, then come back to this page
to read more about the controversy and see if you can resolve it.

Which strategy is better: to stick with your original choice, or to
switch?

This question was asked of Marilyn von Savant, who has a column in a
Sunday magazine section in many newspapers. She replied that it was
better to switch; there was only a 1/3 chance of your original guess
being right, and a 2/3 chance that it was wrong, in which case the
host has shown you how to "switch, and win." According to Marilyn,
your odds of winning are 1 in 3 if you stick with your original
choice, 2 in 3 if you win.

Her column prompted a flood of angry letters, some of them from
mathematicians, telling Marilyn to brush up on her basic
probability. These letters said that the fact that a third door had
been opened to reveal a goat didn't alter the fact that the car was
still equally likely to be behind either of the two remaining closed
doors. According to these letters, your odds of winning are 1 in 2
if you stick with your original choice, and also 1 in 2 if you
switch.

Who is right? Or are neither of them right?

See if you can come up with a mathematical justification for one of
these opinions, and see if you can figure out what's wrong with
the other one.

To help you, you can try
playing the game a few times on our computer
simulation, to gather some experimental evidence. But, because the game
is random, just playing a few times won't tell you much. For example,
flipping a coin 5 times and getting 4 heads is not that unusual
(happens with a 5/32 probability), even though the probability of
getting a head is only 50% not 80%! So you may have to play many times
before your observed proportion of wins gets anywhere close to
the predicted probabilities.

So, to help you get more reliable experimental evidence, you can
have the computer play automatically
times, choosing
each time. (New random data is generated for each game).
Select the number of times you want to play and the strategy you
want the computer to use, then press
.

Once you've come up with the best mathematical explanation you can,
and checked it out against the experimental evidence,
you can move on to the next section to
read a discussion of the mathematics behind the game.